Stable and unstable sets for evolution of parabolic and hyperbolic type
This paper is concerned with the stability and instability of the solutions of the following two initial-boundary value problems: \[ u_t-\Delta u= |u|^{p-1}u,\quad x\in\Omega,\quad t>0,\tag{1} \] \[ u(0,x)= u_0(x),\quad x\in\Omega,\quad u(t,x)= 0,\quad x\in\partial\Omega,\quad t>0, \] and \[ u_{tt}-\Delta u+\delta u_t= |u|^{p-1}u,\quad x\in\Omega,\quad t>0,\tag{2} \] \[ u(0,x)= u_0(x),\quad u_t(0,x)= u_1(t),\quad x\in\Omega,\quad u(t,x)=0,\quad x\in\partial\Omega,\quad t>0. \] The stable set \(W^*\) and the unstable set \(V^*\) are defined by \(W^*= \{u\in H^1_0(\Omega); J(u)<d, I(u)>0\}\cup\{0\}\) and \(V^*= \{u\in H^1_0(\Omega); J(u)<d, I(u)<0\}\), where \[ J(u)={1\over 2} |\nabla u|^2_2- {1\over p+1} |u|^{p+1}_{p+1},\quad I(u)= |\nabla u|^2_2-|u|^{p+1}_{p+1}, \] \[ d=\inf\Biggl\{\sup_{\lambda\neq 0} J(\lambda u);\;0\neq u\in H^1_0(\Omega)\Biggr\}. \] Let \(u\) be the solution of (1) or (2), and \(T_m\) the maximal existence time. The authors show that for the problem (1) \(u(t_0,.)\in W^*\) for some \(t_0\in[0,T_m)\) if and only if \(T_m=+\infty\) and \(\lim_{t\to\infty} |\nabla u(t,.)|_2=0\), and that if either \(u_0\geq 0\) or \(\Omega\) is convex, then \(u(t_0,.)\in V^*\) for some \(t_0\in[0,T_m)\) if and only if \(T_m<+\infty\). Similar results are established for the problem (2) except that the ``only if'' part is shown in the second part. The results are improvements of related results established by many authors, and a different method is used in this paper.
scientific article; zbMATH DE number 986937